Evaluate each expression without using a calculator.
cos (csc⁻¹ (-2))

Verified step by step guidance

1. Understand that \( \csc^{-1}(-2) \) is the angle whose cosecant is \(-2\).
2. Recall that \( \csc(\theta) = \frac{1}{\sin(\theta)} \), so \( \sin(\theta) = -\frac{1}{2} \).
3. Identify the angle \( \theta \) in the range \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \) where \( \sin(\theta) = -\frac{1}{2} \).
4. Recognize that \( \theta = -\frac{\pi}{6} \) satisfies \( \sin(\theta) = -\frac{1}{2} \).
5. Use the identity \( \cos(\theta) = \sqrt{1 - \sin^2(\theta)} \) to find \( \cos(-\frac{\pi}{6}) \).

Key Concepts

Cosecant and Inverse Cosecant

Cosecant is the reciprocal of sine, defined as csc(θ) = 1/sin(θ). The inverse cosecant function, csc⁻¹(x), gives the angle θ such that csc(θ) = x. In this case, csc⁻¹(-2) represents an angle whose cosecant value is -2, which helps in determining the corresponding sine value.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. For the sine function, sin(θ) = opposite/hypotenuse. Knowing the cosecant value allows us to find the sine value, which is crucial for evaluating the cosine of the angle derived from the inverse cosecant.

Cosine Function

The cosine function, cos(θ), relates to the adjacent side and hypotenuse of a right triangle, defined as cos(θ) = adjacent/hypotenuse. Once we determine the sine value from the cosecant, we can use the Pythagorean identity sin²(θ) + cos²(θ) = 1 to find the cosine value needed for the final evaluation.